While we await for the Republicans to decide who will be their candidate for Speaker, it is instructive to look back at the 113th Senate with our new alpha-NOMINATE procedure. Recall that Alpha-NOMINATE is a new form of NOMINATE that is fully Bayesian and is meant to replace W-NOMINATE which is now about 32 years old (the multidimensional version, written by Nolan McCarty and Keith Poole is 24 years old). NOMINATE was designed by Keith Poole and Howard Rosenthal during 1982-1983. It used a random utility model with a Gaussian deterministic utility function (see pages 14 – 15 of the linked 1983 paper) and logistic error (random draws from the log of the inverse exponential). The Gaussian deterministic utility function is able to capture non-voting due to indifference * and* alienation.

Alpha-NOMINATE is a mixture model in which legislators’ utility functions are allowed to be a mixture of the two most commonly assumed utility functions: the quadratic function and the Gaussian function assumed by NOMINATE. The “Alpha” is a parameter estimated by Alpha-NOMINATE that varies from 0 (Quadratic Utility) to 1 (Gaussian Utility). Hence, in one dimension with Alpha = 0, Alpha-NOMINATE is identical to the popular IRT model. Thus Alpha-NOMINATE can actually test whether or not legislators’ utility functions are Quadratic or Gaussian.

Below we apply Alpha-NOMINATE to the 113^{th} Senate. There were 657 total votes in the 113^{th} of which 552 are scalable (at least 2.5% in the minority; that is, votes that are 97-3 to 50-50). We used the R version of Alpha-NOMINATE to perform the analysis. We used 3000 samples from a slice sampler in one dimension with a burn-in of 1000. The first graph shows the Trace and Density plots for alpha.

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The mean of Alpha is 0.9981 with a standard deviation of 0.0018 strongly indicating that the Senators’ utility functions were Gaussian.

The next plot shows the estimated ideal points for the 105 Senators who served during the 113^{th} along with their 95% Credible Intervals. On the left, Presidential candidate Senator Bernie Sanders (I-VT) is located at -2.74 off the left edge of the plot. Near the right end are Presidential candidates Ted Cruz (R-TX) at 1.608 Rand Paul (R-KY) at 1.453. Lindsey Graham is near the left edge of the Republican Party at 1.011.

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There were very few moderates in the 113^{th} Senate. Only Lisa Murkowski (R-AK), Susan Collins (R-ME), and Joe Manchin (D-WV) are between the two parties. Moderation is in short supply in both the Senate and the House. No wonder Republicans are finding it difficult to find someone to be Speaker.